On the Reconstruction of SAS from Other Triangle Congruence Criteria, Part II: Eliminating the Pons Asinorum

In the first part of this work we showed that, within a Hilbert plane deprived of the Side-Angle-Side axiom, the Side-Angle-Angle criterion, together with a ray correspondence principle [$\mathbf{\text{RCT}}$], the existence of angle bisectors [$\mathbf{\text{AB}}$], the congruence of supplements of congruent angles [$\mathbf{\text{SA}}$], and the Pons Asinorum [$\mathbf{\text{PA}}$], suffices to reconstruct SAS. We left open the question of whether [$\mathbf{\text{PA}}$] is genuinely required alongside the other three principles, noting only a qualitative asymmetry in the nature of the principles involved. In this second part we answer this question: we show that \begin{equation*} $\text{SAA}$,\ [$\mathbf{\text{RCT}}$],\ [$\mathbf{\text{AB}}$] ;\vdash; [$\mathbf{\text{PA}}$], \end{equation*} so that [$\mathbf{\text{PA}}$] is redundant among the hypotheses of our main theorem, which improves to \begin{equation*} $\text{SAA}$,\ [$\mathbf{\text{RCT}}$],\ [$\mathbf{\text{AB}}$],\ [$\mathbf{\text{SA}}$] ;\vdash; $\text{SAS}$. \end{equation*} The proof adapts an argument recently given by Donnelly, who reconstructs SAS from SAA together with an angle addition axiom and the existence of angle bisectors.